From ca43e5660ccbd3f4beaaa2073ce7dda05a80eff6 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Mon, 1 Aug 2022 19:58:09 +0200
Subject: Kugel: Introduction and preliminaries (not spherical harmonics, yet)

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+% vim:ts=2 sw=2 et spell tw=78:
+
+\section{Introduction}
+
+This chapter of the book is devoted to the sef of functions called
+\emph{spherical harmonics}. However, before we dive into the topic, we want to
+make a few preliminary remarks to avoid ``upsetting'' a certain type of
+reader. Specifically, we would like to specify that the authors of this
+chapter not mathematicians but engineers, and therefore the text will not be
+always complete with sound proofs after every claim. Instead we will go
+through the topic in a more intuitive way including rigorous proofs only if
+they are enlightening or when they are very short. Where no proofs are given
+we will try to give an intuition for why it is true.
+
+That being said, when talking about spherical harmonics one could start by
+describing their name. The latter may be a cause of some confusion because of
+the misleading translations in other languages. In German the name for this
+set of functions is ``Kugelfunktionen'', which puts the emphasis only on the
+spherical context, whereas the English name ``spherical harmonics'' also
+contains the \emph{harmonic} part hinting at Fourier theories and harmonic
+analysis in general.
+
+The structure of this chapter is organized in the following way. First, we
+will quickly go through some fundamental linear algebra and Fourier theory to
+refresh a few important concepts. In principle, we could have written the
+whole thing starting from a much more abstract level without much preparation,
+but then we would have lost some of the beauty that comes from the
+appreciation of the power of some surprisingly simple ideas. Then once the
+basics are done, we can explore the main topic of spherical harmonics which as
+we will see arises from the eigenfunctions of the Laplacian operator in
+spherical coordinates. Finally, after studying what we think are the most
+beautiful and interesting properties of the spherical harmonics, to conclude
+this journey we will present a few real-world applications, which are of
+course most of interest for engineers.
+
-- 
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